Approximate Solutions for Important Model Systems

Hans Luth

Lucjan Piela

Giuseppe Mussardo

Abstract Chapter 3 discusses the approximation schemes used to approach lattice statistical models that are not exactly solvable. In addition to the mean field approximation, it also considers the Bethe–Peierls approach to the Ising model. Moreover, there is a thorough discussion of the Gaussian model and its spherical version, both of which are two important systems with several points of interest. A chapter appendix provides a detailed analysis of the random walk on different lattices: apart from the importance of the subject on its own, it explains how the random walk is responsible for the critical properties of the spherical model.

Giuseppe Mussardo

Abstract The exact solutions of one-dimensional systems discussed in the previous chapter are particularly simple and mathematically elegant. Unfortunately, they are more an exception than a rule: for higher dimensional lattices, the computation of the partition functions of statistical models usually poses a formidable problem from a mathematical point of view. For that reason, it is of the utmost importance to develop some approximate methods that permit us to analyze the most relevant physical aspects and to extract an estimate of critical exponents or other thermodynamic quantities. In this chapter we will discuss several approximate solutions of the Ising model and its generalizations.

R. Id Betán

The pairing interaction is one of the most important contribution of the residual interaction and then, it is of major importance for the study of many-body systems. One can get solutions of the pairing Hamiltonian throught the Bardeen-Cooper-Schieffer (BCS) or the Lipkin-Nogami (LN) approximations but, the pairing Hamiltonian admit exact solution worked out by Richardson. Nuclei far away from the stability line have important correlations with the continuum part of the energy spectrum, due that the Fermi level is very close to the contiuum thershold. The correlations with the continuum can be included in the many-body description through the complex energy states, called Gamow states. In this work we compare the approximates and exact solutions of the pairing Hamiltonian in real and complex-energy representations. In the application of this formulation to the symmetric Lipkin model, we found that the LN solution is in a good agreement with the exact one; besides, the extension of the BCS solution to the complex energy plane gives solution even for strength below the critical one, which is purely imaginary.